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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク partially ordered ring ： ウィキペディア英語版 partially ordered ring In abstract algebra, a partially ordered ring is a ring (''A'', +, · ), together with a ''compatible partial order'', i.e. a partial order $\backslash leq$ on the underlying set ''A'' that is compatible with the ring operations in the sense that it satisfies: :$x\backslash leq\; y$ implies $x\; +\; z\backslash leq\; y\; +\; z$ and :$0\backslash leq\; x$ and $0\backslash leq\; y$ imply that $0\backslash leq\; x\backslash cdot\; y$ for all $x,\; y,\; z\backslash in\; A$. Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring $(A,\; \backslash leq)$ where $A$'s partially ordered additive group is Archimedean. An ordered ring, also called a totally ordered ring, is a partially ordered ring $(A,\; \backslash leq)$ where $\backslash le$ is additionally a total order.〔〔 An l-ring, or lattice-ordered ring, is a partially ordered ring $(A,\; \backslash leq)$ where $\backslash leq$ is additionally a lattice order. == Properties == The additive group of a partially ordered ring is always a partially ordered group. The set of non-negative elements of a partially ordered ring (the set of elements ''x'' for which $0\backslash leq\; x$, also called the positive cone of the ring) is closed under addition and multiplication, i.e., if ''P'' is the set of non-negative elements of a partially ordered ring, then $P\; +\; P\; \backslash subseteq\; P$, and $P\backslash cdot\; P\; \backslash subseteq\; P$. Furthermore, $P\backslash cap(-P)\; =\; \backslash$. The mapping of the compatible partial order on a ring ''A'' to the set of its non-negative elements is one-to-one;〔 that is, the compatible partial order uniquely determines the set of non-negative elements, and a set of elements uniquely determines the compatible partial order if one exists. If ''S'' is a subset of a ring ''A'', and: # $0\backslash in\; S$ # $S\backslash cap(-S)\; =\; \backslash$ # $S\; +\; S\backslash subseteq\; S$ # $S\backslash cdot\; S\backslash subseteq\; S$ then the relation $\backslash leq$ where $x\backslash leq\; y$ iff $y\; -\; x\backslash in\; S$ defines a compatible partial order on ''A'' (''ie.'' $(A,\; \backslash leq)$ is a partially ordered ring).〔 In any l-ring, the ''absolute value'' $|x|$ of an element ''x'' can be defined to be $x\backslash vee(-x)$, where $x\backslash vee\; y$ denotes the maximal element. For any ''x'' and ''y'', :$|x\backslash cdot\; y|\backslash leq|x|\backslash cdot|y|$ holds. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「partially ordered ring」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.